sections: fix and add a lot of things
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@ -28,6 +28,14 @@ references:
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container-title: Journal of Econometrics
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issued:
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year: 2015
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- type: book
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id: silver86
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author:
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family: Silverman
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given: Bernard W.
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title: Density Estimation for Statistics and Data Analysis
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issued:
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year: 1986
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header-includes: |
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```{=latex}
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@ -177,7 +177,6 @@ $$
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exp(-\x*\x) +
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exp(-(\x - 1.4)*(\x - 1.4)) +
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exp(-(\x - 0.8)*(\x - 0.8)) + 0.1});
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\end{tikzpicture}
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\end{center}
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\setbeamercovered{transparent}
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@ -187,7 +186,7 @@ $$
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## Sample FWHM
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Silverman's rule of thumb:
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Silverman's rule of thumb [@silver86]:
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$$
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\varepsilon = 0.88 \, S_N
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@ -3,14 +3,10 @@
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## KS
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Quantify distance between expected and observed CDF
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. . .
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Quantify distance between expected and observed CDF. KS statistic:
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:::: {.columns}
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::: {.column width=50% .c}
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KS statistic:
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$$
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D_N = \text{sup}_x |F_N(x) - F(x)|
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$$
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@ -22,23 +18,22 @@ Quantify distance between expected and observed CDF
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- sort points in ascending order
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- number of points preceding the point normalized by $N$
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. . .
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:::
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::: {.column width=50%}
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}
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% axes
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\draw [thick, ->] (-2.5,0) -- (0,0) -- (0,4.5);
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\draw [thick, ->] (0,0) -- (2.5,0);
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% empiric
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\draw [cyclamen, fill=cyclamen!20!white] (-2.5,0) rectangle (-1.5,0.5);
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\draw [cyclamen, fill=cyclamen!20!white] (-1.5,0) rectangle (-0.9,1);
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\draw [cyclamen, fill=cyclamen!20!white] (-0.9,0) rectangle (-0.6,1.5);
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\draw [cyclamen, fill=cyclamen!20!white] (-0.6,0) rectangle ( 0.2,2);
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\draw [cyclamen, fill=cyclamen!20!white] ( 0.2,0) rectangle ( 0.5,2.5);
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\draw [cyclamen, fill=cyclamen!20!white] ( 0.5,0) rectangle ( 0.8,3);
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\draw [cyclamen, fill=cyclamen!20!white] ( 0.8,0) rectangle ( 1.6,3.5);
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\draw [cyclamen, fill=cyclamen!20!white] (-0.6,0) rectangle ( 0.5,2);
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\draw [cyclamen, fill=cyclamen!20!white] ( 0.5,0) rectangle ( 0.7,2.5);
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\draw [cyclamen, fill=cyclamen!20!white] ( 0.7,0) rectangle ( 1.2,3);
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\draw [cyclamen, fill=cyclamen!20!white] ( 1.2,0) rectangle ( 1.6,3.5);
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\draw [cyclamen, fill=cyclamen!20!white] ( 1.6,0) rectangle ( 2.3,4);
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\draw [cyclamen, fill=cyclamen!20!white] ( 2.3,0) rectangle ( 2.5,4.5);
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% points
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@ -46,9 +41,9 @@ Quantify distance between expected and observed CDF
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\draw [blue!50!black, fill=blue] (-1.6,-0.1) rectangle (-1.4,0.1); %-1.5
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\draw [blue!50!black, fill=blue] (-1,-0.1) rectangle (-0.8,0.1); %-0.9
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\draw [blue!50!black, fill=blue] (-0.7,-0.1) rectangle (-0.5,0.1); %-0.6
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\draw [blue!50!black, fill=blue] (0.1,-0.1) rectangle (0.3,0.1); % 0.2
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\draw [blue!50!black, fill=blue] (0.4,-0.1) rectangle (0.6,0.1); % 0.5
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\draw [blue!50!black, fill=blue] (0.7,-0.1) rectangle (0.9,0.1); % 0.8
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\draw [blue!50!black, fill=blue] (0.6,-0.1) rectangle (0.8,0.1); % 0.7
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\draw [blue!50!black, fill=blue] (1.1,-0.1) rectangle (1.3,0.1); % 1.2
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\draw [blue!50!black, fill=blue] (1.5,-0.1) rectangle (1.7,0.1); % 1.6
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\draw [blue!50!black, fill=blue] (2.2,-0.1) rectangle (2.4,0.1); % 2.3
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% expected
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@ -56,7 +51,7 @@ Quantify distance between expected and observed CDF
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\draw[domain=-2.5:2.5, yscale=5, smooth, variable=\x, blue, very thick]
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plot ({\x}, {((atan(\x)*pi/180) + pi/2)/pi});
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\pause
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\draw [very thick, cyclamen] (0.8,3.6) -- (0.8,4.05);
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\draw [very thick, cyclamen, <->] (0.5,2.5) -- (0.5,3.25);
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\end{tikzpicture}
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\end{center}
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\setbeamercovered{transparent}
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@ -70,19 +65,17 @@ $H_0$: points sampled according to $F(x)$
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. . .
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If $H_0$ is true:
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If $H_0$ is true: $\sqrt{N}D_N \xrightarrow{N \rightarrow + \infty} K$
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- $\sqrt{N}D_N \xrightarrow{N \rightarrow + \infty} K$
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Kolmogorov distribution with CDF:
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$K$ Kolmogorov variable with CDF:
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$$
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P(K \leqslant K_0) = 1 - p = \frac{\sqrt{2 \pi}}{K_0}
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P(K \leqslant K_0) = \frac{\sqrt{2 \pi}}{K_0}
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\sum_{j = 1}^{+ \infty} e^{-(2j - 1)^2 \pi^2 / 8 K_0^2}
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$$
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. . .
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a $p$-value can be computed
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A $p$-value can be computed
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- At 95% confidence level, $H_0$ cannot be disproved if $p > 0.05$
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@ -1,53 +1,75 @@
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# Trapani test
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## A pathological distribution
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Because of its fat tail:
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\begin{align*}
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\mu_1 &= \text{E}\left[|x|\right] \longrightarrow + \infty \\
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\mu_2 &= \text{E}\left[|x|^2\right] \longrightarrow + \infty
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\end{align*}
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. . .
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No closed form for parameters $\thus$ numerical estimations
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. . .
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For a Moyal PDF:
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\begin{align*}
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E_M[x] &= \mu + \sigma [ \gamma + \ln(2) ] \\
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V_M[x] &= \frac{\pi^2 \sigma^2}{2}
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\end{align*}
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## Infinite moments
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- Check whether a moment is finite or infinite
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\begin{align*}
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\text{infinite} &\thus Landau \\
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\text{finite} &\thus Moyal
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\end{align*}
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. . .
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# Trapani test
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## Trapani test
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::: incremental
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- Random variable $\left\{ x_i \right\}$ sampled from a distribution $f$
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- Sample moments according to $f$ moments
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- Sample moments estimate as $f$ moments
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- $H_0$: $\mu_k \longrightarrow + \infty$
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- Statistic with 1 dof chi-squared distribution
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- Statistic with 1 dof $\chi^2$ distribution
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- $p$-value $\hence$ reject or accept $H_0$
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:::
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## Infinite moments
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- Generate a sample $L$ from a Landau PDF
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- Generate a sample $M$ from a Moyal PDF
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. . .
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\vspace{20pt}
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:::: {.columns}
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::: {.column width=50% .c}
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For the Landau PDF:
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\begin{align*}
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\mu_1 &= \text{E}\left[|x|\right] = + \infty \\
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\mu_2 &= \text{E}\left[|x|^2\right] = + \infty
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\end{align*}
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:::
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::: {.column width=50%}
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. . .
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For the Moyal PDF:
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\begin{align*}
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\mu_1 &= \text{E}\left[|x|\right] < + \infty \\
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\mu_2 &= \text{E}\left[|x|^2\right] < + \infty
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\end{align*}
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:::
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::::
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## Infinite moments
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- Previous tests: points sampled from Landau PDF?
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. . .
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- Trapani test: check whether a moment is finite or infinite
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\begin{align*}
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\text{infinite} &\thus \text{Landau} \\
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\text{finite} &\thus \text{not Landau}
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\end{align*}
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. . .
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- Compatibility test with $\mu_k = + \infty$
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. . .
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- If points were sampled from a Cauchy distribution...
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## Trapani test
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## Trapani test
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- Start with $\left\{ x_i \right\}^N$ and compute $\mu_k$ as:
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@ -61,14 +83,19 @@ For a Moyal PDF:
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$\left\{ a_j \right\}^r$ as:
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$$
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a_j = \sqrt{e^{\mu_k}} \cdot \xi_j
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\thus G'\left( 0, \sqrt{e^{\mu_k}} \right)
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\thus G\left( 0, \sqrt{e^{\mu_k}} \right)
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$$
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. . .
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The greater $\mu^k$, the 'larger' $G'$
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- if $\mu_k \longrightarrow + \infty \thus a_j$ distributed uniformly
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The greater $\mu^k$, the 'larger' $G\left( 0, \sqrt{e^{\mu_k}} \right)$
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$$
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\begin{cases}
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\mu_k \longrightarrow + \infty \\
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r \longrightarrow + \infty
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\end{cases}
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\thus a_j \text{ distributed uniformly}
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$$
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## Trapani test
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@ -115,9 +142,10 @@ The greater $\mu^k$, the 'larger' $G'$
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. . .
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If $a_j$ uniformly distributed and $N \rightarrow + \infty$:
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If $a_j$ uniformly distributed:
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- $\zeta_j (u)$ Bernoulli PDF with $P(\zeta_j (u) = 1) = \frac{1}{2}$
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- $\zeta_j (u)$ Bernoulli PDF with $P\left( \zeta_j (u) = 1 \right) = \frac{1}{2}$
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$\hence E[\zeta_j]_j = \frac{1}{2} \quad \wedge \quad V[\zeta_j]_j = \frac{1}{4}$
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## Trapani test
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@ -130,7 +158,7 @@ $$
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. . .
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If $a_j$ uniformly distributed and $N \rightarrow + \infty$, for the CLT:
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If $a_j$ uniformly distributed, for the CLT:
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$$
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\sum_j \zeta_j (u) \hence
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G \left( \frac{r}{2}, \frac{r}{4} \right)
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@ -151,15 +179,15 @@ $$
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According to L. Trapani [@trapani15]:
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- $r = o(N) \hence r = N^{0.75}$
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- $\underbar{u} = 1 \quad \wedge \quad \bar{u} = 1$
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- $\psi(u) = \chi_{[\underbar{u}, \bar{u}]}$
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- $\underbar{u} = -1 \quad \wedge \quad \bar{u} = 1$
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- $\psi(u) = \frac{1}{\bar{u} - \underbar{u}} \, \chi_{[\underbar{u}, \bar{u}]}$
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. . .
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$\mu_k$ must be scale invariant for $k > 1$:
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$$
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\tilde{\mu_k} = \frac{\mu_k}{ \left( \mu_{\phi} \right)^{k/\phi} }
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\mu_k^* = \frac{\mu_k}{ \left( \mu_{\phi} \right)^{k/\phi} }
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\with \phi \in (0, k)
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$$
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@ -167,7 +195,9 @@ $$
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## Trapani test
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If $\mu_k \ne + \infty \hence \left\{ a_j \right\}$ are not uniformly distributed
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\vspace{20pt}
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Rewriting:
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$$
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\vartheta (u) = \frac{2}{\sqrt{r}}
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@ -178,4 +208,6 @@ $$
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\vspace{20pt}
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. . .
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Residues become very large $\hence$ $p$-values decreases.
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@ -60,22 +60,35 @@ This leads to more different medians:
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\end{align*}
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## Compatibility test
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## Landau Sample
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Comparing results:
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Sample N random points following $L(x)$
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$$
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p = 1 - \text{erf} \left( \frac{t}{\sqrt{2}} \right)\ \with
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t = \frac{|x\ex - x\ob|}{\sqrt{\sigma\ex^2 + \sigma\ob^2}}
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L(x) = \frac{1}{\pi} \int \limits_{0}^{+ \infty}
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dt \, e^{-t \ln(t) -xt} \sin (\pi t)
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$$
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- $x\ex$ and $x\ob$ are the expected and observed values
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- $\sigma\ex$ and $\sigma\ob$ are their absolute errors
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. . .
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At 95% confidence level, the values are compatible if:
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gsl_ran_Landau(gsl_rng)
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## Moyal sample
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Sample N random points following $M_{\mu \sigma}(x)$
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$$
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p > 0.05
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M_{\mu \sigma}(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp
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\left[ - \frac{1}{2} \left(
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\frac{x - \mu}{\sigma}
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+ e^{-\frac{x - \mu}{\sigma}} \right) \right]
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$$
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. . .
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reverse sampling
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- sampling $y$ uniformly in [0, 1] $\hence x = Q_M(y)$
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@ -1,18 +1,62 @@
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# Landau sample
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# Results
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## Sample
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## Compatibility test
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Sample N = 50'000 random points following $L(x)$
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Comparing sample properties:
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$$
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L(x) = \frac{1}{\pi} \int \limits_{0}^{+ \infty}
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dt \, e^{-t \ln(t) -xt} \sin (\pi t)
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p = 1 - \text{erf} \left( \frac{t}{\sqrt{2}} \right)\ \with
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t = \frac{|x\ex - x\ob|}{\sqrt{\sigma\ex^2 + \sigma\ob^2}}
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$$
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- $x\ex$ and $x\ob$ are the expected and observed values
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- $\sigma\ex$ and $\sigma\ob$ are their absolute errors
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. . .
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gsl_ran_Landau(gsl_rng)
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At 95% confidence level, the values are compatible if:
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$$
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p > 0.05
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$$
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## Compatibility test
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\setbeamercovered{}
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\begin{center}
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\begin{tikzpicture}
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%notes
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\draw [very thick, gray] (0,0) -- (0,3);
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\draw [very thick, gray] (-1.45,1.5) -- (1.45,1.5);
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\draw [very thick, gray] (-1.35,1.3) -- (-1.55,1.7);
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\draw [very thick, gray] ( 1.35,1.3) -- ( 1.55,1.7);
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\node [below] at (0,-0.7) {$x\ex$};
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\node [above right] at (1.5,1.5) {$2 \, \sqrt{\sigma\ex^2 + \sigma\ob^2}$};
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% axes
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\draw [very thick, <->] (-5,4) -- (-5,0) -- (5,0);
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% Gaussian
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\draw [domain=-5:5, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {3*exp(-(\x*\x/3))});
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\pause
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% area
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\fill [domain=2:5, smooth, variable=\x, cyclamen!20!white, very thick]
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(2,0) -- plot ({\x}, {3*exp(-(\x*\x/3))}) -- (5,0) -- cycle;
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\fill [domain=-5:-2, smooth, variable=\x, cyclamen!20!white, very thick]
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(-5,0) -- plot ({\x}, {3*exp(-(\x*\x/3))}) -- (-2,0) -- cycle;
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% axes
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\draw [very thick, <->] (-5,4) -- (-5,0) -- (5,0);
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% Gaussian
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\draw [domain=-5:5, smooth, variable=\x, cyclamen, very thick]
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plot ({\x}, {3*exp(-(\x*\x/3))});
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%notes
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\draw [thick, cyclamen] (-2,0) -- (-2,0.8);
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\draw [thick, cyclamen] ( 2,0) -- ( 2,0.8);
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\node at (2,-0.7) {$x\ob$};
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\end{tikzpicture}
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\end{center}
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\setbeamercovered{transparent}
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## Compatibility results:
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@ -71,27 +115,6 @@ FWHM:
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::::
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# Moyal sample
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## Sample
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Sample N = 50'000 random points following $M_{\mu \sigma}(x)$
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$$
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M_{\mu \sigma}(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp
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\left[ - \frac{1}{2} \left(
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\frac{x - \mu}{\sigma}
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+ e^{-\frac{x - \mu}{\sigma}} \right) \right]
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$$
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. . .
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reverse sampling
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- sampling $y$ uniformly in [0, 1] $\hence x = Q_M(y)$
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## Compatibility results:
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Median:
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