89 lines
1.3 KiB
Markdown
89 lines
1.3 KiB
Markdown
# Goal
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## Goal
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Test whether sample comes from a Landau distribution
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. . .
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- sample $L$ from Landau PDF
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- sample $M$ from Moyal PDF
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. . .
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$H_0$: sample from Landau PDF
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- accept $H_0$ for $L$?
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- reject $H_0$ for $M$?
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## Why Moyal?
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Landau and Moyal PDFs are similar
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```{=latex}
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\includegraphics<1>[height=5.5cm]{images/moyal-photo.jpg}
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\includegraphics<2>[height=5.5cm]{images/mondau-photo.jpg}
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\includegraphics<3>[height=5.5cm]{images/landau-photo.jpg}
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```
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## Two similar distributions
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:::: {.columns}
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::: {.column width=50%}
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Landau PDF
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$$
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L(x) = \frac{1}{\pi} \int \limits_{0}^{+ \infty}
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dt \, e^{-t \ln(t) -xt} \sin (\pi t)
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$$
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:::
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::: {.column width=50%}
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Moyal PDF
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$$
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M(x) = \frac{1}{\sqrt{2 \pi}} \exp \left[ - \frac{1}{2}
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\left( x + e^{- x} \right) \right]
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$$
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:::
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::::
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\vspace{1em}
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:::: {.columns}
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::: {.column width=50%}
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:::
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::: {.column width=50%}
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:::
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::::
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## Two similar distributions
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## Statistical tests
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. . .
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- **Properties test**
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compatibility between expected and observed PDF properties
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. . .
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- **Kolmogorov - Smirnov test**
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compatibility between expected and empirical CDF
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. . .
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- **Trapani test**
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test for finite or infinite moments
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